# An introduction to ergodic theory - Walters P.

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(2) If £, < e2 then К) > rn(e2, К).

Definition 7.9. If £ > 0 and К is a compact sublet of X let /■(<:, К, T) = lim sup„^0L, (l//i)logr„(£, K). We write r(e,K,T,d) if we wish to emphasise the metric d.

Remarks

(3) If £t < c2 then г(£ь К, T) > r(e2, К, T) (by Remark 2).

(4) The value of r(e, К, T) could be oo. (An example is given in Remark 14.)

§7.2 Bowen's Definition

169

Definition 7.10. If К is a compact subset of X let h(T; К) = lim__0r(e, К, T). The topological entropy of T is h(T) = supA h{T; K). where the supremum is taken over the collection of all compact subsets of X. We sometimes write hj(T) to emphasise the dependence on J

Before giving any interpretations or explanations of this aefiniiion we shall give an equivalent but “dual” definition. This definition will use the idea of separated sets which is dual to the notion of spanning sets.

Definition 7.11. Let и be a natural number f. > 0 and К be a compact subset of X A subset £ of К is said to be (п,с) separated with respect to T if *, >’ t E, x ф implies cl„(x, y) > e. (i.e., for ,v e E the set Q"=o T~1B(T1x;e) contains no other point of £).

Definition 7.12. If n is a natural number, e > 0 and К is a compact subset of X let sjhi., K) denote the largest cardinality of any (n, c) separated subset of К with respect to T. (We write s„{c, К, T) to emphasise T if we need to.)

Remarks

(5; We have г„(е, К) < s„(c, К) < гп(е/2, К) and hence s„(e, К) < со.

Proof. If E is an (n, f) separated subset of К of maximum cardinality then £ is an (h, e) spanning set for K. Therefore r„(e, K) < s„(e, K). To show the other inequality suppose £ is an (и, s) separated subset of К and F is an (n, c/2) spanning set for K. Define </>:£->£ by choosing, for each x e £, some point ф(х) t F with с!„(х,ф(х)) < e/2. Then ф is injective and therefore the cardinality of E is not greater than that of F. Hence s„(e, K)< r„(c/2. K). □

(6) If sl < e2 then s,l{ElK j > s„(e2,K).

Definition 7.13. If e > 0 and К is a compact subset of X put s(c, К, T) = limsup„_>a) (l//i)log.sn(e,K). We sometimes write s(e,K,T,d) when we need to emphasise the metric d.

Remarks

(7) We have г(е, К. T) < .ф., К, T) < r(sf2, К, T) by Remark (5).

(8) If e, < e2 then s(Eb К, T) > v(a2, К, T).

(9) We have h{T; K) = limr_0 s(e, К, T), by Remark (7), so that h(T) = supKlim£„0.s(£, К, T).

Hence h{l) can be defined using either spanning or separating sets. Remarks

(10) If г'п(е, K) denotes the smallest cardinality of a subset of К that (n.c) spans К then the proof of Remark (5) gives г'„(е,К) < sr(r.,K) < r'„(t:/2, К)

170

7 Topological Lnlropy

so we also have

h(T) = sun lim lim sup - log r'n(n, K).

К £-*0 и-* С/ И

We now make some comments about the definition of h(T).

Remarks

(11) If T is an isometry of (X,d) then clearly dn = d for all ;i so lhat sn(e,K) = s,(e, K) and lid(T) = 0.

(12) For rjr., K) to increase with n the mapping T needs to increase distances between some points. We can think of Л(Т) as a measure of the expansion of T relative to the metric d.

(13) The ideas for the definition come from the work of Kolmogorov on the size of a metric space (see Kolmogorov and Tihomirov [1]). If (Л\ p) is a metric space then a subset F is said to E-span X if V.v e X 3v e F with p(x,y) < e, and a subset E is said to be E-separated if vhenever г :e Г., у Ф r, then p(y,z) > e. The e-entropy of (A", p) is then the logarithm of the minimum number of elements of an E-spanning ^et and the E-capacity is the logarithm of the maximum number of elements in an E-separated sei. So in the above definitions we are considering the metric spaces (h,dn) and г'„(е,К) (see Reirark 10) is the e-entropy of (K,d„) and s„(e, K) is the s-capacity of [K,d„). Therefore

h(T\K) = lim lim sup - (г-entropy of(/C, d„))

E~*0 n~* JO И

= lim lim sup-(e-capacity of (K,dn)).

C~* 0 tl

(14) The following ts an example when r(t, К, T) can be со (see Remark (4)). Consider the real line R with the Euclidean metric and let T(x) — x1. Let К — 13,4]. If x, у > 2 we have d„(x, >’) < e iff jx2""1 — y2" [ < e. By the mean-value theorem |x2”-1 — y2" '| = 2'I-1z2"~1_I|x — y| for some z between x and y, so that dn(x,y) < e implies

Therefore а (и, e)-spanning set for [3,4] contains at least

— 222я-1 — 1 В

points so that

^n- 222"" 1 — 1

rn(e,K) >- and r(e, К, T) = со.

}7.2 Bowrn'b Defimvion

171

We now investigate the dependence of hd(T) on the metric d and then we shall consider hd(T) when X is compact. In this case the definition has a geometric interpretation and we shall show il coincides with the definition

* givsn in §7.1.

I

Definition 7.14. Two metrics d and d' on A arc uniformly equiralenf if id.:(.Y,</) -> (X,d‘) and id.:(AT.rf^ - (X,d) are both uniformly continuous.

In this case, T e UC(X, d) iff T e UC(X, d').

Theorem 7.4. If d and d' are uniformly equivalent and T q bC(X,d) then hd(T> = K(T).

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